Abstract

This paper investigates the oscillatory behavior of the solutions for a three-node neural network with discrete and distributed delays. Two theorems are provided to determine the conditions for oscillating solutions of the model. The criteria for selecting the parameters in this network are derived. Some simulation examples are presented to illustrate the effectiveness of the results.

1. Introduction

Neural networks are complex and large-scale dynamical systems. Time delay is inevitably encountered in implementation of dynamical neural networks and is frequently a source of oscillation and instability. On the one hand, information transmission from one neuron to another neuron may make the response of networks with discrete delays. On the other hand, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, the distributed delays in neural system need to be considered. In the past two decades, many researches have studied various neural networks with discrete and distributed delays [1–4]. In [1], Gopalsamy and Leung studied the following neural network model:

By means of the Lyapunov functional, the authors obtained some necessary and sufficient conditions for the existence of a globally asymptotically stable equilibrium point of system (1). Ruan and Filfil [2] considered a two-neuron network model with multiple discrete and distributed delays as follows:
where the delayed feedback kernel satisfies and ,. Local stability analyses are carried out for model (2). Computer simulations are performed to illustrate the obtained results. Liao et al. [3] discussed a two-neuron system with distributed delays in the frequency domain as follows:
where is a strong kernel. For simplicity, the authors set . Let

By applying the frequency domain approach and analyzing the characteristic equation, the Hopf bifurcation occurs when the mean delay exceeds a critical value. Thus, there is a family of periodic solutions bifurcates from the equilibrium point. Recently, Hajihosseini et al. [4] have investigated the following three-node network model:
where and are parameters. The authors also make the change of variables such that system (6) changes to the following:

Under the restrictive condition , the Hopf bifurcation and stability of the bifurcating periodic solutions have been discussed. There are many authors who have studied the bifurcating periodic solutions for various models [5–14]. It is well known that the basic method in studying the bifurcating periodic solution for a time delay system is to discuss an algebraic equation in which the bifurcating value will be determined. Motivated by the above models we shall concern the existence of oscillating solutions for the following three-node network system with discrete and distributed delays:
where () is a strong kernel function, the passive decay rates , , and are positive constants, , , and ,2, are constants, and delays , , and are nonnegative constants. It was emphasized that bifurcating approach is hard to deal with model (8), because it is very difficult to find the bifurcating parameter when , , and are different nonnegative constants. In order to discuss the existence of oscillating solutions for system (8) we adopt Chafee’s criterion [15]. A time delay system will generate a limit cycle if the system has a unique unstable equilibrium point and bounded solutions. In other words, there exists an oscillatory solution of the model. System (8) can accord with the demands of Chafee’s criterion; we refer the reader to [16, appendix].

2. Preliminaries

First we assume that the activation functions and both are monotone continuous bounded functions. , , , , and for . For example, , and satisfy those conditions. For the kernel function () we have
From (9) we get

Taking the derivative on both sides of system (8) we obtain
Set
so (11) can be rewritten as follows:

Now by setting , , , , , and , we get the following time delay equivalent system of (8):
The linearization of system (15) around the zero point is the following:
System (16) can be written as a matrix form:
where , = , , , , , , , , and . So,
where , , , , , , , , , , , ; , , , , , , , , and .

Proof. Since we assume that and both are monotone continuous bounded functions. So we have and ( and are positive constants). From (8) we obtain
where , , and . Thus, (). This means that the solutions of system (8) are uniformly bounded.

Lemma 2. Assume that the matrix
is a nonsingular matrix. Then system (8) has a unique equilibrium point.

Proof. Noting that and both are monotone continuous bounded functions satisfying . Then , , where and both are higher order infinitesimals when . From (8), an equilibrium point is a solution of the following algebraic equation:
Noting that , then we have
If is another equilibrium point of system (8), neglecting of the higher order infinitesimal, we obtain
The matrix form of (23) is follows:
Since is a nonsingular matrix based on the algebraic knowledge one can have or . This means that system (8) has a unique equilibrium point. Obviously, the unique equilibrium point exactly is zero point since .

3. Oscillating Solutions Analysis

We adopt the following norms of vectors and matrices: , . The measure of the matrix is defined by , which for the chosen norms reduces to .

In order to discuss the instability of equilibrium point for system (8), we consider the equivalent system (15) of (8). Note that the linearized system of (15) is (16). Obviously, if the trivial solution of (16) is unstable, it implies that the trivial solution of system (15) is unstable and thus the instability of the trivial solution of system (8). Therefore, we first have

Theorem 3. Assume that system (8) has a unique equilibrium point and the determinant of matrix is not equal to zero. Let be the eigenvalues of the matrixes and , respectively. If there is at least one > 0 ( may be equal to zero) or there exists one positive real eigenvalue , for some , then the unique equilibrium point, namely, the trivial solution of system (16) is unstable, implying that the equivalent system (15) or (8) generates oscillating solutions.

Proof. Since is a singular matrix, set . For given value of , let be the eigenvalues of the matrix . Thus, the characteristic equation of system (16) is as follows:
where
or
Since there is at least one , without loss of generality, set . From we have the equation . This means that , and there is a Re . So, the unique equilibrium point of system (16) is unstable based on the theory of differential equation. If by corresponding some we have , we pointed out that there exists positive real root for transcendental equation . Let . Then () is a continuous function of . Since , from , we know that . Obviously, there is a suitable large such that since can be suitably small. Thus, by the continuity of (), there is a point, say such that . In other words, there is a positive real eigenvalue. In this case the unique equilibrium point of system (16) is also unstable. Since all solutions of the system are bounded, on the basis of Chafee’s criterion [15], system (15) generates a limit cycle, implying that there exists an oscillating solution of system (15) and therefore system (8).

Theorem 4. Assume that system (8) has a unique equilibrium point and the following inequalities hold:
where and . Then the unique equilibrium point, namely, the trivial solution of system (16) is unstable, implying that the equivalent system (15) or (8) generates oscillating solutions.

Proof. First we consider system (16) in the case and we easily get:
Let ; then for some , () and we have
Consider the scalar equation
with and . According to the comparison theorem of differential equation, one can obtain
We claim that the trivial solution of (31) is unstable. Suppose that this is not true; then the characteristic equation associated with (31) given by
will have a real negative root say such that , where .So we get
Thus
Based on the formula , from (35), we have

Inequality (36) contradicts the first inequality of (28). Therefore, our claim regarding the instability of the trivial solution is valid. Similarly, consider system (16) in the case ; we know that the trivial solution is unstable if the second inequality of (28) holds. Note that . So, the trivial solution of (16) is unstable when condition (28) holds, implying that the trivial solution of (15) is unstable. According to Chafee’s criterion, system (15) generates a limit cycle, suggesting that there is an oscillating solution of system (15) and therefore system (8).

4. Computer Simulations

We use the equivalent system (15) of (8) for computer simulation. In Figure 1 both activation functions and are taken tanh (). Thus, , , and . We select , , , , , , , , and , . Then the eigenvalues of matrix are , , −0.0000, , and −8.2087 and the eigenvalues of matrix are , 0.7560, and . Note that the eigenvalue 0.7560 of matrix is larger than . Based on Theorem 3, the trivial solution of system (15) is unstable. The system generates an oscillating solution. In order to compare the effect of the time delays, we change the delays from (0.2, 0.3, and 0.4) to (1, 1.5, and 2), while keeping the other parameters as shown in Figure 1. We see that the oscillatory amplitude and frequency both are changed (see Figure 2). In Figure 3, we keep the parameter values the same as Figure 2, the activation function , and change the activation function . Then and . We see that the graph is changed slightly in comparison with Figure 2. In Figure 4, we set the activation function and . Then , , , and . Thus and . Note that and . When we select the parameters as follows: , , , , , , , , and , , time delays are (0.5, 0.6, and 0.7). The oscillatory solutions also appeared. This means that the restrictive condition , , and for is only for convenience of the proof.

5. Conclusion

This paper discusses the oscillatory behavior of the solutions for a three-node network model with discrete and distributed delays. Two theorems are provided to ensure the existence of oscillating solutions for the model. Computer simulations suggested that our theorems are only sufficient conditions. How to find a necessary condition is still an open problem.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by Grant no. 11361010 from NNFS of China to Chunhua Feng.